I recently replaced my 1/2 HP well pump with a 1 HP well pump. Why? Because it's what was readily available at the time...

Now, when the pump kicks on, I get a loud bang . I'm thinking I can either 1.) install some type of "start-up pressure absorber" device on the water line, or 2.) install some type of electrical rheostat that eases the amperage into the pump. (# 3 - "put the right pump in" ain't happening. It's 10 degrees outside.)

Your tank is the shock absorber in the system you first asked about. Do you have a bladder tank or a free air control tank. Any way you could have lost air in your tank? Are you still getting fairly long cycles between motor kicking on and off? Is the noise less on startup if you have some heavy usage of water going on like a hose coming directly off the tank. Is the noise the same?

You mentioned you were not interested in replacing it with the “right” pump because of the weather. Is the pump down in the well as a submergible? I don’t see where a larger than needed pump should be causing problems. What you basically will have is a higher flow rate. When you hooked this up was there a large size reduction between the new pump and the old pipe?

Unfortunately, the cost of a VFD controller to slowly ramp up the pump speed would be cost prohibitive especially if the pump will be changed out when warmer weather gets here.
Any loose piping anywhere that could be tightly secured to minimize the kick?

My guess is the new pump just has a higher flow rate and maybe that’s where the line goes into the tank is stepped down causing a partial restriction to the inrush water so as it’s trying to compress the air in the tank it’s now also able to water hammer some. As kok328 pointed out there is a loose pipe that’s smacking against another or something like that. The 1 ½ line coming into the tank goes into what size fitting there?